Two-Dimensional Quon Language: Unifying Cliffords, Matchgates, and Beyond

General quantum states and processes for many-body systems are computationally intractable. However, there exist special classes of quantum circuits that are classically tractable: the Clifford and the matchgate circuits. Each of these classes has proven useful for accomplishing various tasks such as designing fault-tolerant quantum computing protocols and performing variational studies of complex quantum systems. Despite the usefulness of both classes, they appear to be unrelated, as they are characterized by seemingly distinct properties. In this paper, we show that Clifford and matchgate circuits can, in fact, be understood as two distinct special cases of a single underlying structure, for which we present a unifying framework. Specifically, we introduce the 2D Quon diagrammatic language. Our approach employs the combination of Majorana worldlines and their underlying spacetime topology to represent quantum processes and tensor networks. In their full generality, the 2D Quon diagrams are universal, yet they are particularly useful for representing Clifford and matchgate classes. Each of these classes can be efficiently characterized in a pictorially recognizable manner. This capability allows us to push the boundaries of matchgates beyond their standard definition. Furthermore, the pictorial characterization naturally introduces a broader class of efficiently computable tensor network with high non-Cliffordness, high non-matchgateness, and high bipartite entanglement entropy. We discuss various applications of our approach across different disciplines: from understanding well-known results such as the Kramers-Wannier duality and the star-triangle relation of the Ising model, to providing new methods for quantum circuit compilation and variational optimization with novel ansatz states.